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A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.

%I A026424 #66 Apr 10 2025 12:28:16
%S A026424 2,3,5,7,8,11,12,13,17,18,19,20,23,27,28,29,30,31,32,37,41,42,43,44,
%T A026424 45,47,48,50,52,53,59,61,63,66,67,68,70,71,72,73,75,76,78,79,80,83,89,
%U A026424 92,97,98,99,101,102,103,105,107,108,109,110,112
%N A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.
%C A026424 Neither this sequence nor its complement (A028260) contains any infinite arithmetic progression. - _Franklin T. Adams-Watters_, Sep 05 2008
%C A026424 A066829(a(n)) = 1. - _Reinhard Zumkeller_, Jun 26 2009
%C A026424 These numbers can be generated by the sieving process described in A066829. - _Reinhard Zumkeller_, Jul 01 2009
%C A026424 Lexicographically earliest sequence of distinct nonnegative integers with no term being the product of any two not necessarily distinct terms. The equivalent sequence for addition/subtraction is A005408 (the odd numbers), for exponentiation is A259444, and for binary exclusive OR is A000069. - _Peter Munn_, Mar 16 2018
%C A026424 The equivalent lexicographically earliest sequence with no term being the product of any two distinct terms is A026416. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - _Peter Munn_, Mar 16 2019
%H A026424 T. D. Noe, <a href="/A026424/b026424.txt">Table of n, a(n) for n = 1..10000</a>
%H A026424 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.
%H A026424 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%H A026424 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a> - _Reinhard Zumkeller_, Jul 01 2009
%F A026424 Sum 1/a(n)^m = (zeta(m)^2-zeta(2m))/(2*zeta(m)), Dirichlet g.f. of A066829. - Ramanujan.
%F A026424 n>=2 is in sequence if n is not the product of two smaller elements. - _David W. Wilson_, May 06 2005
%F A026424 A001222(a(n)) mod 2 = 1. - _Reinhard Zumkeller_, Oct 05 2011
%F A026424 Union of A000040, A014612, A014614, A046308 etc. - _R. J. Mathar_, Jul 09 2012
%p A026424 isA026424 := proc(n)
%p A026424     if type(numtheory[bigomega](n) ,'odd') then
%p A026424         true;
%p A026424     else
%p A026424         false;
%p A026424     end if;
%p A026424 end proc:
%p A026424 A026424 := proc(n)
%p A026424     option remember;
%p A026424     if n =1 then
%p A026424         2;
%p A026424     else
%p A026424         for a from procname(n-1)+1 do
%p A026424             if isA026424(a) then
%p A026424                 return a;
%p A026424             end if;
%p A026424         end do:
%p A026424     end if;
%p A026424 end proc: # _R. J. Mathar_, May 25 2017
%t A026424 Select[Range[2, 112], OddQ[Total[FactorInteger[#]][[2]]] &] (* _T. D. Noe_, May 07 2011 *)
%t A026424 (* From version 7 on *) Select[Range[2, 112], LiouvilleLambda[#] == -1 &] (* _Jean-François Alcover_, Aug 19 2013 *)
%t A026424 Select[Range[150],OddQ[PrimeOmega[#]]&] (* _Harvey P. Dale_, Oct 04 2024 *)
%o A026424 (Haskell)
%o A026424 a026424 n = a026424_list !! (n-1)
%o A026424 a026424_list = filter (odd . a001222) [1..]
%o A026424 -- _Reinhard Zumkeller_, Oct 05 2011
%o A026424 (PARI) is(n)=bigomega(n)%2 \\ _Charles R Greathouse IV_, Sep 16 2015
%o A026424 (Python)
%o A026424 from math import isqrt, prod
%o A026424 from sympy import primerange, integer_nthroot, primepi
%o A026424 def A026424(n):
%o A026424     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
%o A026424     def f(x): return int(n+1+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,m)) for m in range(2,x.bit_length()+1,2)))
%o A026424     m, k = n, f(n)
%o A026424     while m != k: m, k = k, f(k)
%o A026424     return m # _Chai Wah Wu_, Apr 10 2025
%Y A026424 Cf. A008836, A028260 (complement).
%Y A026424 Apart from initial term, same as A026422.
%Y A026424 Cf. A026416 and cross-references therein.
%Y A026424 Cf. A000028, A000069, A005408, A259444.
%K A026424 nonn,easy,nice
%O A026424 1,1
%A A026424 _N. J. A. Sloane_